How To Solve Second Order Linear Homogeneous Differential Equation | Involving Complex Roots

To solve any Second Order Linear Homogeneous Differential Equation, first this you need to do, is to transform the equation in to an auxiliary or characteristics equation in the form: ar²+be+c=0
We have already seen how to do that in our previous lesson.

The next move is to solve for r which are the roots of the equation (r intercept).
Then determine the nature of the roots and substitute in to the following equations, depending on the nature of roots.

y=C₁eʳ¹ˣ+C₂eʳ²ˣ. when you obtain real and distinct roots

y=(C₁+C₂x)eʳˣ when you Obtain real and equal roots.

and finally, if you Obtain a complex solution in the form: r= m+si or r = m-si
where i is imaginary number, and m and s are real numbers, then
y=eᵐˣ[C₁cos(st)+C₂sin(st)]

[georgesadler]

Professor Tambuwal, thank you for an exceptional video/lecture on Solving Second Order Linear Homogeneous Differential Equations Involving Complex Roots. Please clean up the initial part of the problem by letting y equal to e raised to the r x and then proceed to the final solution.

[R_]

Thank you so much for explaining it so nicely!

[OyebisiOladejo]

Thank you for the fantastic explanation

[NEWAFRICAMOVIES]

Gaskiya tambuwa ka burgeni sosai hakan yayimin Allah yataimaka

[UbaidKhan-cffz]

Why we considered ( beta as positive 2 )?

[ليدبرواآياته-مح]

Assalamu malam, ina buqatar na iya maths, dun babu abunda natsana kamarsa, naga kuma Allah ya maka baiwarsa, ina buqatar ka aza ni ahanya ko nadace, Allah yaqara daraja da albarka.